ΦH: [Σ∞+G/H,Σ∞+G/H]G∗ −→[Σ∞+W(H),Σ∞+W(H)]W∗ (H)
is a retraction and thus in particular surjective. Further, Proposition 6.5.1 implies that ΦH ◦proj∗ = 0.
Hence, it remains to show that the map
proj∗: [Σ∞+G/H, GnH Σ∞+EP[H]]G∗ −→[Σ∞+G/H,Σ∞+G/H]G∗
is injective and Ker ΦH ⊂ Im(proj∗). For this we choose a set {g} of double coset representatives forH\G/H. By Corollary 6.5.3, there is a commutative diagram with all vertical arrows isomorphisms
[Σ∞+G/H, GnH Σ∞+EP[H]]G∗
=
proj∗ //[Σ∞+G/H,Σ∞+G/H]G∗
∼=
[Σ∞+G/H, GnH Σ∞+EP[H]]G∗
∼=
(GnHproj)∗ //[Σ∞+G/H, GnH S]G∗
∼=
L
[g]∈H\G/H[S,Σ∞+ ResgH∩HgH(c∗g(EP[H]))]H∗ ∩gH
L
[g]∈H\G/H
(proj)∗
//L
[g]∈H\G/H[S,S]H∩∗ gH. We will now identify the summands of the lower horizontal map. For this one has to consider two cases:
Case 1. H∩gH=H: In this case ResgH∩HgH(c∗g(EP[H])) =c∗g(EP[H]) is a model for classifying space of P[H] (and hence G-homotopy equivalent to EP[H])). By Proposition 6.4.1, we get a short exact sequence
0 //[S,Σ∞+ ResgH∩HgH(c∗g(EP[H]))]H∩∗ gH proj∗//[S,S]H∩∗ gHΦH∩g H//[S,S]∗ //0.
Case 2. H∩gH is a proper subgroup of H: In this case ResgHH∩gH(c∗g(EP[H])) is an (H∩gH)-contractible cofibrant (H∩gH)-space and hence the map
[S,Σ∞+ ResgH∩HgH(c∗g(EP[H]))]H∗ ∩gH proj∗//[S,S]H∩∗ gH is an isomorphism.
Altogether, after combining the latter diagram with Case 1. and Case 2., we see that the map
proj∗: [Σ∞+G/H, GnHΣ∞+EP[H]]G∗ −→[Σ∞+G/H,Σ∞+G/H]G∗
is injective. It still remains to check that Ker ΦH ⊂Im(proj∗). For this, first note that H∩gH=Hif and only ifg∈N(H). Further, ifg∈N(H), then the double coset class HgH is equal togH. Hence, the set of those double cosets [g]∈H\G/H for which the equalityH∩gH =H holds is in bijection with the Weyl group W(H). Consequently, using the latter diagram and Case 1. and Case 2., one gets an isomorphism
[Σ∞+G/H,Σ∞+G/H]G∗/Im(proj∗)∼= M
W(H)
[S,S]∗∼= [Σ∞+W(H),Σ∞+W(H)]W∗ (H). On the other hand, we have already checked that
ΦH: [Σ∞+G/H,Σ∞+G/H]G∗ −→[Σ∞+W(H),Σ∞+W(H)]W∗ (H)
is surjective and this yields an isomorphism
[Σ∞+G/H,Σ∞+G/H]G∗/Ker ΦH ∼= [Σ∞+W(H),Σ∞+W(H)]W∗ (H).
Combining this with the previous isomorphism implies that the graded abelian group [Σ∞+G/H,Σ∞+G/H]G∗/Im(proj∗) is isomorphic to [Σ∞+G/H,Σ∞+G/H]G∗/Ker ΦH. Now if the grading ∗ >0, then [Σ∞+G/H,Σ∞+G/H]G∗ is finite and it follows that Im(proj∗) and Ker ΦH are finite groups of the same cardinality (Subsection 2.7). Since we already know that Im(proj∗) ⊂Ker ΦH (We have already observed that this is a consequence of Proposition 6.5.1.), one finally gets the equality Im(proj∗) = Ker ΦH. For ∗= 0 a Five lemma argument completes the proof. We do not give here the details of the case
∗= 0 as it is irrelevant for our proof of Theorem 3.1.3.
7 Proof of the main theorem
In this section we complete the proof of Theorem 3.1.3 and hence of Theorem 1.1.1.
We start by recalling from [MM02, IV.6] the F-model structure on the category of G-equivariant orthogonal spectra, where F is a family of subgroups of a finite group G.
7.1 The F-model structure and localizing subcategory determined by F Let Gbe a finite group andF a family of subgroups of G.
Definition 7.1.1. A morphism f:X −→ Y of G-equivariant orthogonal spectra is called anF-equivalence if it induces isomorphisms
f∗:π∗HX ∼= //πH∗ Y
onH-equivariant homotopy groups for anyH ∈F. Similarly, a morphismg:X −→Y in Ho(SpOG) is called an F-equivalence if it induces an isomorphism on π∗H for any H ∈F.
The category ofG-equivariant orthogonal spectra has a stable model structure with weak equivalences theF-equivalences and with cofibrations theF-cofibrations [MM02, IV.6.5]. By restricting our attention to those orbitsG/H which satisfyH ∈F, we can obtain the generatingF-cofibrations and acyclicF-cofibrations in a similar way as for the absolute case of SpOG [MM02, III.4] (see Subsection 2.5 and Subsection 2.6). We will denote this model category by SpOG,F.
AnyF-equivalence can be detected in terms of geometric fixed points. To see this we need the following proposition which relates the classifying spaceEF with the concept of an F-equivalence:
Proposition 7.1.2 ([MM02, IV.6.7]). A morphism f: X −→ Y of G-equivariant or-thogonal spectra is an F-equivalence if and only if 1∧f:EF+∧X−→EF+∧Y is a G-equivalence, i.e., a stable equivalence of orthogonal G-spectra.
Corollary 7.1.3. A morphism f:X −→Y of G-equivariant orthogonal spectra is an F-equivalence if and only if for any H∈F, the induced map
ΦH(ResGH(f)) : ΦH(ResGH(X))−→ΦH(ResGH(Y))
on H-geometric fixed points is a stable equivalence of (non-equivariant) spectra.
Proof. By Proposition 7.1.2, f:X −→Y is anF-equivalence if and only if 1∧f:EF+∧X−→EF+∧Y
is a stable equivalence of orthogonalG-spectra. But the latter is the case if and only if ΦH(ResGH(1∧f)) : ΦH(ResGH(EF+∧X))−→ΦH(ResGH(EF+∧Y))
is a stable equivalence of spectra for any subgroup H ≤G ([May96, XVI.6.4]). Now using that the restriction and geometric fixed points commute with smash products as well as the defining properties of EF, we obtain the desired result.
By definition ofF-equivalences andF-cofibrations we get a Quillen adjunction id : SpOG,F oo //SpOG : id.
After deriving this Quillen adjunction one obtains an adjunction L: Ho(SpOG,F)oo //Ho(SpOG) :R
on the homotopy level. We now examine the essential image of the left adjoint functor L. Since a weak equivalence in SpOG is also a weak equivalence in SpOG,F, the unit
id−→RL
of the adjunction (L,R) is an isomorphism of functors. Hence the functor L: Ho(SpOG,F)−→Ho(SpOG)
is fully faithful.
Proposition 7.1.4. For any X∈SpOG,F, there are natural isomorphisms L(X)∼=EF+∧LX∼=EF+∧X.
Proof. LetλX:Xc−→X be a (functorial) cofibrant replacement ofX in SpOG,F. By [MM02, Theorem IV.6.10], the projection mapEF+∧Xc−→Xcis a weak equivalence in SpOG. On the other hand, Proposition 7.1.2 implies that the morphism of G-spectra 1∧λX:EF+∧Xc −→ EF+∧X is a weak equivalence in SpOG. This completes the proof.
Next, note that the triangulated category Ho(SpOG,F) is compactly generated with {Σ∞+G/H |H∈F}
as a set of compact generators. Indeed, this follows from the following chain of isomor-phisms:
[Σ∞+G/H, X]Ho(Sp
O G,F)
∗ ∼= [EF+∧Σ∞+G/H, EF+∧X]G∗ ∼= [Σ∞+G/H, EF+∧X]G∗ ∼=πH∗ (EF+∧X)∼=π∗HX.
The first isomorphism in this chain follows from Proposition 7.1.4 and from the fact that L is fully faithful. The second isomorphism holds since H ∈F. Finally, the last isomorphism is an immediate consequence of Corollary 7.1.3.
Proposition 7.1.5. The essential image of the functor L: Ho(SpOG,F) −→ Ho(SpOG) is exactly the localizing subcategory generated by {Σ∞+G/H|H∈F}.
Proof. The functor L is exact and as we already noted, Ho(SpOG,F) is generated by the set {Σ∞+G/H |H ∈F}. Next, by Proposition 7.1.4, for any H∈F,
L(Σ∞+G/H)∼=EF+∧Σ∞+G/H.
The projection map EF+∧Σ∞+G/H −→Σ∞+G/H is a weak equivalence in SpOG. The rest follows from the fact that Lis full.
Next, we need the following simple lemma from category theory.
Lemma 7.1.6. Let
L:D oo //E :R.
be an adjunction and assume that the unit id−→RL
is an isomorphism (or, equivalently, L is fully faithful). Further, suppose we are given morphisms
X α //Z oo β Y
in E such that X and Y are in the essential image of L and R(α) and R(β) are isomorphisms in D. Then there is an isomorphism γ:X ∼= //Y in E such that the diagram
X γ //
α@@@@@
@@ Y
~~~~~~β~
Z commutes.
Proof. One has the commutative diagram LR(X)
counit
∼=
LR(α)
∼= //LR(Z)
counit
LR(Y)
LR(β)
∼=
oo
counit
∼=
X α //Zoo β Y,
where the left and right vertical arrows are isomorphisms since X and Y are in the essential image of L and the functor L is fully faithful. We can choose γ:X ∼= //Y to be the composite
X counit
−1 //LR(X) LR(α) //LR(Z) (LR(β))
−1 //LR(Y) counit //Y.
Corollary 7.1.7. Let F be a family of subgroups of G and suppose X and Y are in the essential image of L: Ho(SpOG,F)−→Ho(SpOG) (which is the localizing subcategory generated by {Σ∞+G/H | H ∈ F} according to 7.1.5). Further assume that we have maps
X α //Z oo β Y
such thatπH∗ α andπH∗ β are isomorphisms for any H ∈F (Or, in other words,α and β are F-equivalences.). Then there is an isomorphism γ:X ∼= //Y such that the diagram
X γ //
α@@@@@
@@ Y
~~~~~~β~
Z commutes.
Proof. We apply the previous lemma to the adjunction L: Ho(SpOG,F)oo //Ho(SpOG) :R and use the isomorphismπ∗HR(T)∼=πH∗ T,H ∈F.
7.2 Inductive strategy and preservation of induced classifying spaces